A theorem on the existence of trace-form generalized entropies

Abstract

An analytic technique is proposed, which allows to generate many new examples of entropic functionals generalizing the standard Boltzmann–Gibbs entropy. Our approach is based on the existence of a group-theoretical structure, which is intimately related with the notion of entropy, as clarified in recent work of the author. The new entropies proposed satisfy the first three Shannon–Khinchin axioms and are composable (at least in a weak sense). By combining them, multi-parametric examples of entropies can be realized. As a by-product of the theory, entropic functionals related to the Riemann zeta function and other number-theoretical functions are introduced.